Periodic In-plane Loads Research Articles

The nonlinear instability of transverse vibration behavior of functionally graded materials sandwich rectangular plates of under a harmonic loading and subsonic flow

This work is devoted to studying the combined action of periodic in-plane load and tangential subsonic flow on the nonlinear mechanism of the out-plane deformation of the functionally graded composite plate. Three distribution functions are proposed to investigate the distribution of the transverse shear strains and stresses across the thickness of the plates. By using Hamilton’s variational principle, the equations governing the plate instability boundaries are formulated. A nonlinear differential equation describing the first mode of the plate dynamic instability behavior by employing Galerkin’s method. The method of multiple time scales is used to obtain a periodic one-mode solution, which directly leads to the solvability condition. The different resonance cases for the interaction between the frequency of out- plane deformation and the external forces are discussed to discover the extent of the sandwich functionally graded material (FGM) rectangular plate resistance in the presence of external influences. Various bifurcation diagrams for different cases of resonance are obtained versus the basic parameters. In view of this study, we could get a deeper look at the complexity of frequency components in resonance responses of the sandwich FGM rectangular plate. The results could provide some suggestions for the sandwich plate’s vibration reduction design or fault diagnosis field or design fault diagnosis.

Free Vibration and Instability Analysis of Sandwich Plates with Carbon Nanotubes-Reinforced Composite Faces and Honeycomb Core

In this study, the instability regions of a honeycomb sandwich plate are investigated for different end conditions under periodic in-plane loading. The core layer of the sandwich plate is made of carbon nanotube (CNT)/glass fiber-reinforced honeycomb and the face layers of CNT/glass fiber- reinforced laminated composite. The governing equations are derived using classical laminated plate theory (CLPT) and solved numerically by using finite element formulation. The effectiveness of the developed finite element formulation is demonstrated by comparing the results in terms of natural frequencies with those available in the literature. The effects of CNT wt.% on the core material, CNT wt.% on the skin material, ply orientation and various end conditions on the variation of natural frequencies, loss factors and instability regions are studied. Finally, some inferences for the effects of CNT reinforcement on the honeycomb sandwich plate subjected to the periodic in-plane loads are discussed.

Dynamic Instability Analysis and Design Charts of Curved Panels with Linearly Varying Periodic In-Plane Load

The present paper reports an extensive study on dynamic instability characteristics of curved panels under linearly varying in-plane periodic loading employing finite element formulation with a quadratic isoparametric eight nodded element. At first, the influences of three types of linearly varying in-plane periodic edge loads (triangular, trapezoidal and uniform loads), three types of curved panels (cylindrical, spherical and hyperbolic) and six boundary conditions on excitation frequency and instability region are investigated. Further, the effects of varied parameters, such as shallowness parameter, span to thickness ratio, aspect ratio, and Poisson’s ratio, on the dynamic instability characteristics of curved panels with clamped–clamped–clamped–clamped (CCCC) and simply supported-free-simply supported-free (SFSF) boundary conditions under triangular load are studied. It is found that the above parameters influence significantly on the excitation frequency, at which the dynamic instability initiates, and the width of dynamic instability region (DIR). In addition, a comparative study is also made to find the influences of the various in-plane periodic loads, such as uniform, triangular, parabolic, patch and concentrated load, on the dynamic instability behavior of cylindrical, spherical and hyperbolic panels. Finally, typical design charts showing DIRs in non-dimensional forms are also developed to obtain the excitation frequency and instability region of various frequently used isotropic clamped spherical panels of any dimension, any type of linearly varying in-plane load and any isotropic material directly from these charts without the use of any commercially available finite element software or any developed complex model.

Parametric instability analysis of laminated composite plate subject to various types of non-uniform periodic in-plane edge load

In this paper, the dynamic instability behavior of laminated composite plate structure is predicted under different types of non-uniform harmonic edge compressive loading. A nine degree of freedom (DOF) type polynomial based higher order shear deformation theory (HSDT) is considered for the finite element discretization of the plate. The application of non-uniform harmonic in-plane edge load causes the in-plane stress variation to be non-uniform. Hence, the in-plane stresses need to be evaluated, prior to the instability analysis, for the evaluation of critical buckling load. These in-plane stresses are computed using in-plane stress analysis approach using finite element method. The differential equations of motion of the system are turned into a set of ordinary differential equations (Mathieu type equations) and solved as a general eigenvalue problem as per the procedure suggested by Bolotin. The accuracy and flexibility of the present model are validated by contrasting the present outcomes and the available solution. Further, the impact various parameters like span-thickness ratio, aspect ratio, diverse in edge constraints, different types of non-uniform periodic edge load, etc. on the dynamic instability behavior of the laminated composite plate are contemplated.

Stability behavior of variable stiffness composite panels under periodic in-plane shear and compression

Dynamic stability characteristics of variable stiffness composite plates under periodic in-plane loads are investigated here using a shear-deformable sixteen-node plate bending element. The modal transformation technique is used to reduce the equations of motion for the flexural vibration problem of a rectangular panel into a set of Mathieu-Hill equations. The diagonal and off-diagonal dominance of the parametric coefficient matrix is studied for the representative cases of in-plane loading, i.e., periodic compression/shear in presence or absence of static compression for the isotropic and composite plates. The multi-modal response (participating and accompanying modes) of the shear panels under periodic edge traction is demonstrated through the time history analysis. Then, the method of multiple scales is used to identify the dynamic instability regions corresponding to the simple (single mode flexural vibration) or combination (two mode flexural vibrations) type of resonances in various (isotropic and laminated composite made of straight or curvilinear fibers) shear panels, for which results are not available in the literature.

Nonlinear dynamic instability analysis of laminated composite thin plates subjected to periodic in-plane loads

In this paper, the dynamic instability of thin laminated composite plates subjected to harmonic in-plane loading is studied based on nonlinear analysis. The equations of motion of the plate are developed using von Karman-type of plate equation including geometric nonlinearity. The nonlinear large deflection plate equations of motion are solved by using Galerkin’s technique that leads to a system of nonlinear Mathieu-Hill equations. Dynamically unstable regions, and both stable- and unstable-solution amplitudes of the steady-state vibrations are obtained by applying the Bolotin’s method. The nonlinear dynamic stability characteristics of both antisymmetric and symmetric cross-ply laminates with different lamination schemes are examined. A detailed parametric study is conducted to examine and compare the effects of the orthotropy, magnitude of both tensile and compressive longitudinal loads, aspect ratios of the plate including length-to-width and length-to-thickness ratios, and in-plane transverse wave number on the parametric resonance particularly the steady-state vibrations amplitude. The present results show good agreement with that available in the literature.

Dynamic instability analysis of rotating delaminated tapered composite plates subjected to periodic in-plane loading

In this study, the dynamic instability analysis of rotating delaminated thickness tapered composite plates with different taper configurations subjected to periodic in-plane loads is performed. The governing differential equations of motion of the various configurations of a rotating thickness mid-plane delaminated tapered laminated composite plate are presented in the finite element formulation based on classical laminated plate theory including various rotational effects. The finite element formulation developed for the dynamic instability analysis of rotating delaminated tapered composite plate is validated by comparing the parametric exciting frequencies evaluated using the present finite element method (FEM) with those presented in available literature and comparing the natural frequencies of rotating delaminated tapered composite plate evaluated using the present FEM with experimentally measured results. Various parametric studies are also performed to study the effect of taper configurations, rotational speed, delamination length, delamination location and in-plane periodic loads on stability of the structure. It was seen that the influences of rotating speed on the principal DIR spectra among taper configurations and uniform plate are significant and the normalized width of principal and second instability regions of parametric resonance frequencies increase in the order of uniform, TC3, TC2 and TC1 for all the rotating speeds.

Dynamic instability of delaminated composite skew plates subjected to combined static and dynamic loads based on HSDT

This study deals with the effects that various parameters have on the dynamic stability of delaminated composite skew structures under various periodic in-plane loads, based on the higher-order shear deformation theory (HSDT). The upper and lower boundaries of the instability regions are determined using the method suggested by Bolotin. The numerical results obtained for square and skew plates without delamination are in good agreement with those reported by other researchers. The behavior of laminated skew plate structures with various delamination sizes was also investigated to study the interactions between the skew angle and other various parameters such as, the delamination size and the fiber angle of each layer. The effect of the magnitude of static and dynamic in-plane loads on the dynamic instability region is also analyzed.

Parametric instability of delaminated composite plates subjected to periodic in-plane loading

This paper deals with the study of the parametric instability characteristics of delaminated composite plates subjected to periodic in-plane load. A first order shear deformation theory based on finite element method is developed for studying the instability region of mid-plane delaminated composite plate. A detailed parametric investigation is carried out to study the influence of the delamination area, number of layers, degree of orthotropy, aspect ratio and static in-plane load on the dynamic stability characteristics of delaminated cross-ply plates. The results of vibration and buckling of delaminated composite plates are compared with previous numerical studies in the literature and the authors’ experimental investigations. The boundaries of instability regions are obtained by using Bolotin’s method and are represented in the non-dimensional load amplitude-excitation frequency plane. It is observed that the natural frequencies and the critical buckling load of the plates decrease with an increase in delamination size. The increase in delamination size also causes dynamic instability regions to be shifted to lower excitation frequencies.

Dynamic Stability of Superelliptical Plates Resting on Elastic Foundations under Periodic In-Plane Loads

AbstractThe dynamic instability of thin, clamped, flat isotropic elastic panels of superelliptical planform set on a two-parameter Winkler-Pasternak foundation and subjected to uniformly distributed pulsating in-plane loads is theoretically investigated. The plate perimeter is described by a superelliptic function with a power corresponding to shapes ranging from a circle (ellipse) to a square (rectangle). The classical Galerkin procedure is used to reduce the problem into a set of nondimensional coupled Mathieu-Hill equations, and the regions of parametric instability for principal and combination resonant frequencies are determined by applying Hsu’s technique. The effects of the superellipticity parameter, plate aspect ratio, and foundation stiffness on the dynamic stability behavior (onset and width of the instability regions) are examined. Moreover, the transient flexural response of the panel under static in-plane load and subjected to a transverse uniform step force is obtained. The validity of the ...

PARAMETRIC INSTABILITY OF LAMINATED COMPOSITE CYLINDRICAL PANELS SUBJECTED TO PERIODIC NON-UNIFORM IN-PLANE LOADS

In the present investigation, the dynamic instability regions of shear deformable cross-ply laminated and composite cylindrical panels subjected to periodic nonuniform in-plane loads are reported. Since the applied in-plane load is nonuniform, initially the static part of the nonuniform in-plane loads are applied and the stresses (σx, σy and τxy) within the panel are evaluated by the solution of cylindrical panel membrane problem. Subsequently, superposing the stress distribution due to static and dynamic in-plane loads, the stress distributions within the panel are obtained. Using these stress distributions the governing equations of the problem are derived through Hamilton's variational principle based on higher-order shear deformation theory of elastic shell theory incorporating von Kármán-type nonlinear strain displacement relations. The governing partial differential equations are reduced into a set of ordinary differential equations (Mathieu-type of equations) by employing Galerkin's method. The instability boundaries of Mathieu equation corresponding to periodic solutions of period T and 2T are determined using Fourier series. Effect of various parameters like static and dynamic load factors, aspect ratio, thickness-to-radius ratio, shallowness ratio, linearly varying in-plane load, parabolic in-plane load and various boundary conditions on the instability regions are investigated.

DYNAMIC STABILITY OF LAMINATED COMPOSITE PLATES WITH PIEZOELECTRIC LAYERS SUBJECTED TO PERIODIC IN-PLANE LOAD

In this paper, the dynamic stability characteristics of laminated composite plates with piezoelectric layers subjected to periodic in-plane load are studied. The finite element method is employed using a modified first-order shear deformation plate theory (MFSDT). The formulation includes the effects of transverse shear, in-plane, and rotary inertia. The boundaries of dynamic instability regions are obtained using Bolotin's approach. The structural system is considered to be undamped. The correctness of the formulation is established by comparing the authors' results with those available in the published literature. The effects of control voltage, static buckling load parameter, number of stacking layers, and thickness of plate on the principal and second instability regions are investigated for cross-ply laminated composite plate.

Nonlinear dynamic stability of composite plates with piezoelectric layers subjected to periodic in-plane load

Nonlinear dynamic stability characteristics of composite plates subjected to periodic in-plane load are investigated via the finite element method with dynamic response analysis. Piezoelectric actuator layers are embedded at the top and bottom of the laminated composite plate. The theoretical formulation is based on the modified first order shear deformation theory (MFSDT) incorporating the von Karman type nonlinear strains. The structural system is considered to be undamped. The nonlinear governing equations obtained are solved using the Newmark's direct integration method coupled with the direct iteration method. The boundaries of dynamic instability regions are obtained using Bolotin's approach. Effects of in-plane forcing frequency and applied voltage on the characteristic features of dynamic stability behaviour are investigated using both linear and nonlinear dynamic response analyses.

Finite element dynamic stability analysis of laminated composite skew plates containing cutouts based on HSDT

The finite element dynamic stability analysis of laminated composite skew structures subjected to in-plane pulsating forces is carried out based on the higher-order shear deformation theory (HSDT). The two boundaries of the instability regions are determined using the method proposed by Bolotin. The numerical results obtained for square and skew plates with or without central cutout are in good agreement with those reported by other investigators. The new results for laminated skew plate structures containing cutout in this study mainly show the effect of the interactions between the skew angle and other various parameters, for example, cutout size, the fiber angle of layer and thickness-to-length ratio. The effect of the magnitude of the periodic in-plane load on the dynamic instability index is also investigated.

Nonlinear vibration and dynamic stability analysis of composite plates

Large amplitude flexural vibration characteristics of composite plates under transverse harmonic pressure or periodic in-plane load are investigated here using the shear deformable finite element method. The nonlinear stiffness matrix is formulated based on von Kármán's assumptions to obtain the stiffness interaction between the in-plane and bending degrees of freedom. Further, the flexural motion of the plate is assumed to be harmonic and the in-plane movement is assumed to be periodic. The nonlinear matrix amplitude equation is obtained by employing Galerkin's method. The coupled nonlinear matrix amplitude equation (in-plane motion is coupled with flexural motion) is solved to obtain (1) nonlinear free flexural vibration frequencies of isotropic and composite plates with different in-plane boundary conditions, (2) flexural vibration amplitudes of such plates under transverse harmonic pressure or periodic in-plane load. Finally, the time history analysis is carried out to understand the steady-state or unsteady nature of the flexural vibration under different loading and boundary condition.

Dynamic Stability Behavior of 3D Braided Composite Plates Integrated with Piezoelectric Layers

This article presents a dynamic stability analysis of a simply supported 3D braided composite laminated plate with surface-bonded piezoelectric layers, subjected to electrical and periodic in-plane mechanical loads. The braided composite plate is produced by the four-step 1 × 1 technique. It is assumed that the yarn is transversely isotropic, and the matrix is isotropic. A fiber inclination model is used to predict the effective stiffness matrices of the braided composite laminates. Theoretical formulations are based on Reddy’s higher order shear deformation plate theory and include piezoelectric effects. Double Fourier series is employed to convert the dynamic governing equations into a linear system of Mathieu—Hill equations from which the boundary points on the unstable regions are determined by Bolotin’s method. Numerical illustrations show the influence of fiber volume fraction, braiding angle, inclination angle, static load level, applied voltage, and aspect ratio on the dynamic stability, buckling and free vibration of the braided composite plate.

Parametric instability of delaminated composite spherical shells subjected to in-plane pulsating forces

Abstract The dynamic stability analysis of delaminated spherical shell structures subjected to in-plane pulsating forces is carried out based on the higher-order shell theory of Sanders. In the finite element (FE) formulation, the seven degrees of freedom per each node are used with transformations in order to fit the displacement continuity conditions at the delamination region. The boundaries of the instability regions are determined using the method proposed by Bolotin. The numerical results obtained for plates and shells are in good agreement with those reported by other investigators. The new results for delaminated shell structures in this study mainly show the effect of the interactions between the radius–length ratio and other various parameters, for example, delamination size, the fiber angle of layer and location of delamination in the layer direction. The effect of the magnitude of the periodic in-plane load on the instability regions is also investigated.

CrossRef Listing of Deleted DOIs

Based on Timoshenko-Mindlin kinematic hypotheses and Hamilton’sprinciple,a dynamic non-linear theory for general laminated circular cylindrical shellswith transverse shear deformation is developed.A multi-mode solution for periodic in-plane loads is formulated for the non-linear dynamic stability of an anti-symmetricangle-ply cylinder with its ends elastically restrained against rotation.The resultedequations in terms of time function are solved by the incremental harmonic balancemethod.

Dynamic stability analysis of composite skew plates subjected to periodic in-plane load

Here, the dynamic stability characteristics of simply supported laminated composite skew plates subjected to a periodic in-plane load are investigated using the finite element approach. The formulation includes the effects of transverse shear deformation, in-plane and rotary inertia. The boundaries of the instability regions are obtained using the Bolotin's method and are represented in the non-dimensional load amplitude-excitation frequency plane. The principal and second instability regions are identified for different parameters such as skew angle, thickness-to-span ratio, fiber orientation and static in-plane load.

Analysis of dynamic instability for arbitrarily laminated skew plates

The dynamic instability and nonlinear response of rectangular and skew laminated plates subjected to periodic in-plane load are studied. Based on von Karman plate theory, the large amplitude dynamic equations of thin laminated plates are derived by applying the approach of generalized double Fourier series. On the assumed mode shape, the governing equations are reduced to the Mathieu equation using Galerkin's method. The incremental harmonic balance (IHB) method is applied to solve the nonlinear temporal equation of motion, and the region of dynamic instability is determined in this work. Calculations are carried out for isotropic, angle-ply and arbitrarily laminated plates under two cases of boundary conditions. The principal region of dynamic instability associated with the effect of the stacking sequence of lamination and the skew angle of plate are also investigated and discussed. The results obtained indicated that the instability behavior of the system is determined by the several parameters, such as the boundary condition, number of the layers, stacking sequence, in-plane load, aspect ratio, amplitude and the skew angle of plate.